Desperately seeking birthday mates! or what Maths teachers get up to on Saturday nights!

Neal, Denise ; Muir, Tracey ; Manuel, Kate 等

Saturday night. Lygon Street, Carlton. Nine mathematics teachers and teacher educators sharing a Thai banquet after a day working on a national numeracy project. What could possibly go wrong?

Can you imagine a group of nine mathematics educators out for a quiet dinner in a beautiful Thai restaurant in Lygon Street, Melbourne after a busy day of work on a Saturday? Sounds like a boring event, hey?

Well, not this group! After spending a day planning for the 2013 Reach for the Stars event (just how much rubbish do we recycle?), the working conversation continued over dinner. We were pretty pleased with our efforts with developing ideas for 2013, and so turned our attention to possible themes for 2014. The conversation shifted to birthdays:

"Wouldn't it be a great idea if next year's Reach for the Stars event focussed on birthdays?"

"By the way, did you know that there only needs to be quite a small number of people in a room for there to be a good chance that there are two people with a birthday on the same day?" suggested one of the group.

"Isn't that called the Birthday Paradox?" asked someone at the other end of the table.

"How do we find out about that?" said another.

With that, it was good old Google to the rescue! A search on the phone quickly found the Birthday Paradox, which indicated that there only needs to be 23 people in a room for there to be a 50% chance of two people with a birthday on the same day, and only 57 people to have a 99% probability of a match.

What would a group of mathematics teachers do on a Saturday night but engage in some data collection and statistical investigation? We started with those at our table--'When is your birthday?"--and started accumulating dates: 29 November, 8 October, 8 May. Everyone shared, but alas no matches! Perhaps this was not surprising, as there were only nine of us, but we were not prepared to give up our quest. We were determined to find a match.

Looking about the restaurant, we estimated that there were about 60 people in the room; surely there would be a match if the information on Wikipedia and all that we knew about this problem was right! One of the group took the initiative and set about asking everyone seated in the restaurant, systematically collecting the data on a serviette--but still no matches.

We would not be defeated! A rather extroverted member of the group called over the elegant young Thai waitress and sent her to the kitchen with specific instructions to ask the entire kitchen staff for the dates of their birthdays and to write down the dates on a serviette. Dutifully she returned with all of us eagerly waiting.

Could we believe it? There were still no matches!

We still did not give up! Having paid the bill, the next step was to move our data collection activities out into Lygon Street. We met some very interesting people, including a table of men wearing brightly coloured fake fur hats that resembled bears and other unusual animals, a group of men and women smoking hookahs on the street, and lots of groups happily enjoying dinner. Each group was accosted by our 'leader', who introduced us with these words: "We are maths teachers and we are conducting an important survey. We need to know when your birthday is as we are trying to find two birthdays that match." Amazingly, no-one said, "No". They actually took us quite seriously and told us their birthdays, which we added to the ever-growing list of dates on our serviette. We thought that there might be a skewed sample, in that it was Saturday night (4 May), and we logically expected that there would be a few people out celebrating birthdays, but this did not seem to be the case. There was much laughter as we continued our quest!

Finally we had a match: two birthdays on 17 May! It had taken much longer than expected and many more than 57 people. Much laughter and fun was had by all, proving that maths can be fun, that statistics can be collected in many places and used for a range of purposes, that sometimes reality can go against what we think we should expect from the theoretical probability, and that really rare events do happen.

The maths behind the birthday problem

The best (only?) way to determine the likelihood of a birthday match within a group of people actually involves thinking about the opposite (complementary) problem: what are the chances of having no matches in the group? If we can find out the probability of having no matches at all in a group, then the probability of finding a match will be 1 minus this probability, since 'having one or more matches in the group' is the complementary event for 'having no matches at all'.

To find out the probability of having a group with no matches, we will imagine trying to create a room full of people, no two of whom share a birthday. To begin with, imagine a big room with just one person in it. In rather unimaginative fashion we will call this individual Person-1. With Person-1 being the only person in the room, it is obvious there are no 'twin' birthdays in the room.

Now suppose that a new person, Person-2, comes to the door. Person-2 is only allowed into the room if he or she does not share a birthday with Person-1. What are the chances of that happening? Ignoring 29 February (to keep things simple), the chances are

364/365

as Person-1 has used up one of the birthdays in the year. This means that the chance of having two people in the room who do not share a birthday is

364/365.

We can now deduce that when there are two people the chance of having a match--which is the 'opposite' or 'complement' of not having a match--is the complement of the 'no matches' probability, or

1 - [364/365].

Not surprisingly, for two people this turns out to be

[1/365].

Things will, however, get more complicated from now on!

Let us keep going. Imagine that Person-3 turns up. This person is only allowed into the 'no matches' room if his or her birthday does not match either of the two people already in the room. Since those two people have used up two birthdays from the year, the chance of being able to let Person-3 into the room must be

363/365.

From this we can conclude that the chances of having three people in the room without a common birthday must be

[364/365] x [365/363]

since we need to have a non-matching second person and then a third person who does not match either of Person-1 or Person-2. The complement of this,

1 - ([364/365] x [363/365])

then gives the probability of the three people in the room actually having a matching pair of birthdays (or even more than one), since

P(at least one match) = 1 - P(no matches).

If Person-4 turns up, the probability that he or she will be allowed into the 'twin-free' room is

362/365

and so the chances of having a four-person 'no matches' room is

[364/365] x [363/365] x [362/365].

This implies that the chances of having at least one birthday match with four people is

1 - ([364/365] x [363/365] x [362/365]).

If we continue our argument, allowing people into the room only if they do not share a birthday with anyone already in there, then you can see that it becomes progressively less likely that a person will be allowed to enter because there are more and more birthdays already in the room and any new arrival cannot match up with any of these. Thus, the chance that Person-k, the kth person to arrive, will be allowed into the room is

[365 - k + 1]/365.

The resulting probability for being able to make a room full of k people, none of whom share a birthday, is thus

[364/365] x [363/365] x [362/365] x ... x [[365 - k + 2]/365] x [[365 - k + 1]/365]

(there are k - 1 numbers in that big long product).

Our complement argument says that the event of having at least one match is the complement of having no matches at all, and so

P(at least one match) = 1 - P(no matches),

and so we can now conclude--drum roll--that the chances of having one or more birthday matches in a room of k people is

1 - ([364/365] x [363/365] x [362/365] x ... x [[365 - k + 2]/365] x [[365 - k + 1]/365]).

We can set up a spreadsheet to work out the values of this probability for different values of k. The results are shown in Figure 1. A graph of these probabilities as a function of group size is shown in Figure 2. As can be seen, 23 people are needed in order for the probability of having a shared birthday in the group to be at least 50%, but it then continues to climb. Once there are 40 people in the group, the chance of a shared birthday is almost 90%. With 83 people we are very close to 100%. If our mad Lygon Street adventurers had to ask 70 people (or even more--they lost count!) before getting a match, then they experienced a very rare event indeed: there is a greater than 99% chance of getting a match with 69 people, and, in fact, the chance of not getting a match with so many people is only about 0.1% or 0.001. Of course, once you have 366 people (or 367 if you want to allow for 29 February birthdays) a match is guaranteed, but, until then, there is a small (possibly very small) chance that you will have a twin-free group--it just could be very unlikely (unless you happen to be mad maths diners on Lygon Street)!

[FIGURE 1 OMITTED]

[FIGURE 2 OMITTED]

Denise Neal

Department of Education, Tasmania

<denise.neal@education.tas.gov.au>

Tracey Muir

University of Tasmania

<tracey.muir@utas.edu.au>

Kate Manuel

The Australian Association of Mathematics Teachers

<kmanuel@aamt.edu.au>

Sharyn Livy

Mathematical Association of Victoria

<slivy@mav.vic.edu.au>

Helen Chick (1)

University of Tasmania

<helen.chick@utas.edu.au>

(1) Helen has signed an affidavit to declare that she was nowhere near Lygon Street on the night in question. The remaining authors have no alibis for the date and time of the events recounted here.